Abstract
We present an extended polynomial remainder sequence algorithm XPRS for R[X] whereR is a domain. From this we derive a Berlekamp-Massey algorithm BM/R overR. We show that if (α) is a linear recurring sequence in a factorial domainU, then the characteristic polynomials for (α) form aprincipal ideal which is generated by a primitive minimal polynomial. Moreover, this generator ismonic when U[[X]] is factorial (for example, whenU is Z orK[X 1,X2,...,Xn] whereK is a field). From XPRS we derive an algorithm MINPOL for determining the minimal polynomial of (α) when an upper bound on the degree of some characteristic polynomial and sufficiently many initial terms of (α) are known. We also show how to obtain a Berlekamp-Massey type minimal polynomial algorithm from BM/U and state BM_MINPOL/K explicitly with a further refinement. Examples are given forU=Z, GF(2)[Y].
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Fitzpatrick, P., Norton, G.H. The Berlekamp-Massey algorithm and linear recurring sequences over a factorial domain. AAECC 6, 309–323 (1995). https://doi.org/10.1007/BF01235722
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DOI: https://doi.org/10.1007/BF01235722